ACARA v9 CONTENT DESCRIPTION “experiment with the effects of the variation of parameters on graphs of related functions, using digital tools, making connections between graphical and algebraic representations, and generalising emerging patterns”
Builds on: Quadratic Functions and Equations (AC9M9A04). This unit builds on quadratic functions and straight-line graphs, connecting their algebraic forms to the shapes they produce. That two-way fluency between symbol and shape is the lasting skill this unit builds, and it underpins the graphing you will meet in measurement, statistics and beyond.
A function and its family
A function like y equals x squared is not a lonely curve but the head of a whole family. Change one of the numbers in its rule, a parameter, and the graph moves, stretches, or flips in a way that is entirely predictable once you have seen it happen a few times. This unit is about running those experiments, ideally with graphing software where you can drag a slider and watch the curve respond, and then putting the pattern into words that hold for the whole family. The real prize is the link between the algebra and the picture: every change to the equation has a matching change in the graph.
A family from one curve
Change one number in the rule and the same basic parabola reappears in a new position: a whole family from one shape.
All three curves are the very same shape as y = x²; adding or subtracting a number only slides the curve, producing a family of related functions.
Adding a constant shifts up or down
Begin with the gentlest parameter, a constant added to a squared function. Compare y equals x squared with y equals x squared plus two and y equals x squared minus three. Every output simply rises by two or falls by three, so the entire parabola slides vertically without changing shape. The vertex moves from the origin to zero comma two, then to zero comma minus three. The rule is clean: adding k lifts the graph up by k, and subtracting moves it down. The number k is a vertical translation.
Sliding up and down (k)
Adding k lifts every point by k and subtracting lowers it, so the vertex rides straight up or down the y-axis.
The vertex moves from (0, 0) to (0, 2) and (0, −3): adding k slides the whole graph up by k, subtracting slides it down. The shape never changes.
The horizontal shift and its reversed sign
Horizontal shifts are the famous trap, because the sign seems backwards. The graph of y equals x minus three all squared is the basic parabola moved three units to the right, with its vertex at three comma zero, even though the rule contains a minus three. Likewise x plus two all squared moves two units to the left. The reason is that the curve bottoms out where the bracket equals zero, and x minus three is zero when x is three. So inside the bracket, minus h shifts right by h.
Sliding left and right (h)
The bracket is zero at x equals h, so the curve bottoms out there: minus h inside shifts the graph right, and the sign looks reversed.
The trap: y = (x − 3)² shifts RIGHT by 3, because the bracket is zero at x = 3. The sign reverses, so x + 2 inside shifts LEFT by 2.
Vertex form: where algebra meets the graph
Pairing this with the vertical shift gives the vertex form y equals x minus h all squared plus k, whose vertex sits exactly at h comma k. This vertex form is where algebra and graph shake hands. Take y equals x minus one all squared minus four. Reading off the parameters, the vertex is at one comma minus four. Expanding the same expression gives x squared minus two x minus three, which is precisely the quadratic graphed earlier in this year with roots at minus one and three. Two different-looking rules describe one identical curve, and being able to move between the factored, expanded and vertex forms is what fluency in quadratics really means.
Vertex form meets expanded form
Vertex form reads the turning point straight off the rule; expanding the same expression reveals the roots. Two rules, one curve.
Reading vertex form gives the vertex (1, −4); expanding gives x² − 2x − 3 with roots −1 and 3. Two different-looking rules, one identical curve.
The coefficient that stretches and flips
A third parameter changes the steepness. In y equals a times x squared, the number a stretches the curve vertically. With a equal to two the parabola is narrower, climbing twice as fast, so at x equals two the height is eight rather than four. With a equal to a half it is wider and flatter. Most strikingly, a negative value of a flips the parabola upside down: y equals minus x squared opens downward, turning the lowest point into a highest point. One parameter controls both how sharp the curve is and which way it opens.
Stretch and flip (a)
The coefficient a is a single dial for both steepness and direction: bigger is narrower, between zero and one is wider, negative flips it over.
With a = 2 the curve is narrower, with a = 0.5 wider, and with a = −1 it reflects to open DOWNWARD. One coefficient controls both steepness and direction.
The same two dials on a straight line
Straight lines tell the same story with their own two parameters. In y equals m x plus c, the gradient m sets the steepness and direction, so a larger m tilts the line more steeply and a negative m makes it fall from left to right. The constant c is simply the y-intercept, the height at which the line crosses the vertical axis. Comparing y equals two x plus one with y equals two x plus four shows two parallel lines, identical in slope but lifted apart by the change in c, exactly mirroring how k shifted the parabola.
The same dials on a line
A straight line has its own two parameters: the gradient m sets steepness and direction, the constant c sets the height it crosses the axis.
y = 2x + 1 and y = 2x + 4 are parallel, lifted apart by c; y = −x + 5 has a different gradient. m is the steepness dial, c the y-intercept dial.
Generalising: every parameter is a dial
Across all these families one general lesson emerges. A parameter is a dial: turning it produces a specific, repeatable transformation of the graph, and the same kind of dial behaves the same way wherever it appears. Adding a constant shifts a graph, a coefficient stretches or reflects it, and a number inside a bracket shifts it horizontally with the sign reversed. Spotting these patterns means you can predict a graph from its equation, and sketch an equation from its graph, without plotting a single extra point.
Quick self-check
1. How does the graph of y = x² + 2 differ from y = x²?
2. In which direction does the graph of y = (x − 3)² move compared with y = x²?
3. What is the vertex of y = (x − 1)² − 4?
4. What does a negative value of a do to the graph of y = a x²?
5. Lines y = 2x + 1 and y = 2x + 4 are related how?